We show that given a fixed irrational rotation of the $d$-dimensional
torus, any analytic SL(2, R)-cocycle can be perturbed in such a way that the
Lyapunov exponent becomes positive. This result strengthens and generalizes
previous results of Krikorian [6] and Fayad-Krikorian [5]. The
key technique is the analyticity of $m$-functions (under the hypothesis of
stability of zero Lyapunov exponents), first observed and used in the
solution of the Ten-Martini Problem [2].